Brownian Dynamics Simulation of Protein-Protein ...METHODS: A Companion to Methods in Enzymology 14,...

METHODS: A Companion to Methods in Enzymology 14, 329–341 (1998)

Article No. ME980588

Brownian Dynamics Simulation of Protein–ProteinDiffusional EncounterRazif R. Gabdoulline and Rebecca C. Wade1

Structural Biology Programme, European Molecular Biology Laboratory, Meyerhofstrasse 1, 69117Heidelberg, Germany

tion rate of the two proteins is defined by their diffu-Protein association events are ubiquitous in biological sys- sional encounter (3). The determination of whether a

tems. Some protein associations and subsequent responses are reaction or response is diffusion controlled is clearlydiffusion controlled in vivo. Hence, it is important to be able to crucial to protein design studies.compute bimolecular diffusional association rates for proteins. Recently, there has been quite a dramatic increaseThe Brownian dynamics simulation methodology may be used to in the number of experimental measurements of thesimulate protein–protein encounter, compute association rates, kinetics of protein–protein interactions (4). This is inand examine their dependence on protein mutation and the na- part due to the recent availability of instruments basedture of the physical environment (e.g., as a function of ionic on surface plasmon resonance (4, 5), which, in princi-strength or viscosity). Here, the theory for Brownian dynamics ple, provide a generally applicable noninvasive methodsimulations is described, and important methodological as-

to measure the kinetics of protein–protein interactionspects, particularly pertaining to the correct modeling of electro-without the necessity for labeling. Experimentallystatic forces and definition of encounter complex formation, aremeasured association rates cover a wide range fromhighlighted. To illustrate application of the method, simulations103 to 109 M01 s01. The lower limit is dictated largelyof the diffusional encounter of the extracellular ribonuclease,by experimental limits on the rates measurable (4),barnase, and its intracellular inhibitor, barstar, are described.while the upper limit is based on measured associationThis shows how experimental rates for a series of mutants andrates in solution for the association of thrombin andthe dependence of rates on ionic strength can be reproducedhirudin (6) and of barnase and barstar (7). The follow-well by Brownian dynamics simulations. Potential future uses ofing properties of protein association rates indicate dif-the Brownian dynamics method for investigating protein–protein

association are discussed. q 1998 Academic Press fusion control:

a. Fast, i.e., ¢ 106 M01 s01 under typical conditions.b. Inverse dependence on solvent viscosity (1) and,

therefore, linear dependence on the proteins’ relativediffusion constant.Protein–protein association is fundamental to pro-

c. Dependence on the ionic strength of the solution,cesses such as signal transduction, transcription, cellindicating the importance of long-range electrostaticcycle regulation, and immune response. The speed offorces.the association phase puts an upper limit on the speed

d. Temperature dependence following the tempera-of the response resulting from protein binding in vivo.ture dependence of the solvent viscosity. The diffusionFrequently, at least one of the interacting proteins isconstants of proteins in water approximately doublefree to move in the intra- or extracellular environmentover the temperature range 288–313 K (8) , and, asand must find its binding partner by diffusion. Thehas been observed for some antibody–antigen com-association rate is limited by the time required to bringplexes (9, 10), this causes diffusion-controlled rates tothe ‘‘reactive patches’’ of the proteins together by diffu-double too.sion (1, 2) so that the proteins form an ‘‘encounter com-

e. Sensitivity to the diffusional environment, e.g.,plex.’’ When the postdiffusional step of the associationwhether, before complexation, both proteins are free toprocess is much faster than diffusional dissociation, thediffuse in solution or one of them is immobilized on a‘‘reaction’’ is diffusion controlled; that is, the associa-surface.

None of these five properties alone or in combination1 To whom correspondence should be addressed. Fax: /49 6221387 517. E-mail: [email protected]. is a proof of diffusional control. Therefore, detailed sim-

3291046-2023/98 $25.00Copyright q 1998 by Academic PressAll rights of reproduction in any form reserved.

AID Methods 0588 / 6208$$$121 03-23-98 10:05:53 metha AP: Methods

330 GABDOULLINE AND WADE

ulation of diffusional association is a necessary step which lead to the apparently random motion of theparticles that is diffusion. Such motion was first re-in defining the mechanism of interaction. Brownian

dynamics (BD) simulations provide a means to compute corded in 1827 by Brown, who observed erratic motionsof pollen grains in water. Einstein (15) and von Smolu-bimolecular diffusional association rates . They have

been used for calculating protein–protein and en- chowski (16) showed that the displacement Dr of aparticle undergoing Brownian motion in time Dt iszyme–substrate association rates using both atomic-

detail models and highly simplified models [for re- given byviews, see (11, 12)]. BD simulations provide insightsinto the properties and mechanisms of diffusion-con- »Dr2… Å 6DDt, [1]trolled association and have been used for the designof protein mutants with altered association rates [see,

where D is the translational diffusion coefficient of thee.g., (13)].particle andBD simulations by Northrup and Erickson (14) of

the association of model spherical proteins showed thatD Å kbT/6pha, [2]associations rates in the absence of long-range attrac-

tive forces should be Ç2 1 106 M01 s01. This is about1000-fold faster than would be expected solely on the where kb is Boltzmann constant, T is absolute tempera-basis of geometric criteria for bringing the ‘‘reactive ture, h is solvent viscosity, and a is the hydrodynamicpatches’’ on the proteins into contact. However, in a radius of the particle.diffusional system the particles do not rebound away The dynamics of diffusional motion are described byfrom each other after colliding with a nonreactive con- the Langevin equation and there are a number of waystact as in a Newtonian system; instead, they diffuse to solve this equation (17). In simulations of protein–close to each other, making multiple collisions and rota- protein association, the method used is that presentedtionally reorienting between them. This diffusive en- by Ermak and McCammon in 1978 for simulation oftrapment effect results in fast association rates of the the motion of Brownian particles (18). In this method,order 106 M01 s01, which are only about 1000-fold a trajectory is generated as a set of snapshots of theslower than the theoretical diffusion-limited Smolu- particles at time intervals of Dt. The positions of thechowski rate for two isotropically reactive spheres. particles are computed from the Ermak–McCammonRates of Ç106 M01 s01 are often seen for protein–pro- equations for translational and rotational motion. Intein complexation, e.g., antibody–antigen association. their simplest forms, these are as follows:However, rates exceeding 106 M01 s01 and ranging up

• The translational displacement Dr of each particleto Ç109 M01 s01 are observed for some protein–proteinduring each time step is given byencounters, indicating that electrostatic interactions

facilitate binding in these cases. Electrostatic steeringis the main focus of BD simulations of protein–protein Dr Å (kbT)01DFDt / R, [3]association.

In this paper, we first present the theory for simulat- where F is the systematic force on the particle beforeing protein diffusion by BD and computing bimolecular the step is taken, and R is a random vector satisfyingassociation rates. We then describe how simulations »R… Å 0 and »R2… Å 6DDt. The (kbT)01D factor by whichare performed, highlighting the most important techni- F is multiplied models the damping effect of solventcal aspects. We illustrate this with a case study of the friction.association of barnase and barstar. We conclude by out- • The rotational displacement angle Dw of each par-lining other systems to which either the method has ticle at each time step is given bybeen applied or that provide possible subjects of futurestudy.

Dw Å (kbT)01DRTDt / W, [4]

where T is the torque acting on the particle before theTHEORYstep is taken, DR is the rotational diffusion constant ofthe particle, and W is a random rotation angle satis-

Simulation of Diffusional Motion by Brownian Dynamics fying »W… Å 0 and »W2… Å 6DRDt.For simulations of protein–protein interactions inThe theory of Brownian motion describes the dy-

namic behavior of particles, the mass and size of which which the proteins are treated as rigid bodies, thereare only two solute particles. Therefore, translationalare larger than those of the molecules of the solvent in

which they are immersed. These particles are subject motion is simulated for one of the proteins (protein II)relative to the position of the other (protein I) (see Fig.to stochastic collisions with the solvent molecules


331BROWNIAN DYNAMICS SIMULATION OF DIFFUSIONAL ENCOUNTER

1). The displacement of protein II is given by Eq. [3], interactions, and nonuniform reactivity. Nevertheless,‘‘boundary conditions’’ given by an analytical solutionwith D replaced by the relative translational diffusion

constant. to the diffusion equation are necessary to use the re-sults of BD simulations to compute rates. Thus, theThe effects of hydrodynamic interactions are usually

neglected for simulations of protein–protein associa- association rate k computed from a BD simulation isgiven bytion but can be treated by substituting tensors for the

diffusion constants in the above equations.To use these equations to propagate diffusional mo- k Å kD(b)b`, [5]

tion, the time step, Dt, must be small enough that theforces and torques on the particles (and the gradient where kD(b) is the steady-state rate constant for twoof the diffusion tensors) remain effectively constant particles approaching within a distance b (see Fig. 1)during the time step. On the other hand, the motion and b` is the probability that having reached thiscan be analyzed only over periods exceeding a particle’s separation, the particles will go on to ‘‘react’’ andmomentum relaxation time, i.e., Dt @ mD/kbT, where form an encounter complex rather than diffuse apartm is the mass of the particle. to infinite separation. kD(b) must be computed ana-

lytically, whereas b` is computed from BD simula-Computation of Bimolecular Diffusional Associationtions in which thousands of trajectories are gener-Ratesated, each of which is started with the particlesComputation of bimolecular association rates re- separated by distance b and is truncated when theyquires solution of the diffusion equation. This can be reach a separation distance q.solved analytically only for systems with simple geome- For two spherical particles with relative diffusiontry and charge distribution. Numerical BD simulations constant D and no interparticle forces, kD(b) is givenare necessary to permit evaluation of the effects on rate by the Smoluchowski expression (19):constants of a protein’s complex shape, heterogeneous

charge distribution, internal motions, hydrodynamickD(b) Å 4pDb. [6]

When interparticle forces are present,

kD(b) Å 4pF*`bSe{E(r)/kBT}r2D DdrG

01

, [7]

where E(r) is the centrosymmetric interaction energybetween the particles, which depends on their separa-tion distance r.b` is usually given by (20)

b` Å b[1 0(1 0 b)V], [8]

where V Å kD(b)/kD(q) and b is the fraction of trajector-ies in which encounter complex formation occurs beforethe particles diffuse to a separation distance of q. Themultiplier for b in Eq. [8] is a correction factor to ac-count for the truncation of trajectories at a finite sepa-ration distance, q.FIG. 1. Schematic diagram indicating the setup of the system for

BD simulations to compute bimolecular diffusion-controlled rate con- Several alternative ways to compute associationstants. Simulations are conducted in coordinates defined relative to rates using BD simulations have been developed to im-the position of the central protein (protein I). At the beginning of prove efficiency, e.g., the WEBDS (Weighted-Ensembleeach trajectory, the second protein is positioned with a randomly

Brownian Dynamics Simulations) method (21), an algo-chosen orientation at a randomly chosen point on the surface of therithm to compute time-dependent rate coefficients (22),inner sphere of radius b. BD simulation is then performed until this

protein diffuses outside the outer sphere of radius q. During the and an algorithm to remove the outer sphere by usingsimulation, satisfaction of reaction criteria for encounter complex an ‘‘m surface’’ (23). These methods have been appliedformation is monitored. The radius b is chosen so that outside the either to simulation of simple model systems for pro-sphere the forces between the proteins are centrosymmetric and in-

tein–protein association (24) or to atomic-detail simu-side they are anisotropic. The radius q is chosen so that the inwardreactive flux at separation q is centrosymmetric (22). lations of enzyme–substrate association (25). They



have not yet been applied to simulations of protein– the proteins are assigned lower dielectric constants,typically 2–4. The boundary between these dielectricsprotein association using detailed protein models.is assigned from atomic coordinates and radii. While

Modeling of Forces Governing Diffusional Motion of there are a number of numerical methods for solvingProteins the Poisson–Boltzmann equation, the most commonly

used is the finite-difference method in which a set ofFor simulations of protein–ligand association, inter-difference equations of the following type are solvedmolecular forces and torques are given by the sum ofiteratively (27):electrostatic and exclusion forces. Short-range attrac-

tive interactions such as hydrogen bonding and van derWaals interactions are not modeled because they are h2 ∑ eij(fi 0 fj)

h/ cih3eik2fi Å 4pqi. [10]

not as important for encounter complex formation asfor formation of the final bound complex and because

Here, j runs over the six grid points adjacent to gridit is too computationally demanding to model them overpoint i, eij is the permittivity of the face connectingthe time scales of BD simulations. These time scalesi and j, ci is the fraction of the volume accessible toare orders of magnitude longer than those for molecu-counterions, ei is the permittivity at point i, and qi islar dynamics simulations in which a more completethe charge enclosed. The parameters of this grid willmodel of forces is used. Additional forces that are some-influence the accuracy of the calculation (see below).times modeled for BD simulation of bimolecular associ-

The finite-difference method can be used to computeations involving proteins are hydrodynamic interac-the intermolecular free energy, and hence the intermo-tions and intramolecular forces to model internallecular forces, for a system consisting of two low dielec-flexibility.tric solutes in a high-dielectric solvent. This is, how-ever, too computationally demanding for use in BDElectrostatic Forcessimulations for which forces are required at every timeA range of electrostatic models have been employedstep in thousands of trajectories. Instead, the testin BD simulations of protein–ligand association. A con-charge approximation has usually been adopted. Fortinuum model of the solvent is adopted for the BD simu-this, the electrostatic potential of one protein (proteinlations and it is also used for computing electrostaticI) is computed on a grid. The second protein (proteinforces. The electrostatic models, however, vary in theII) then moves on the potential grid of protein I andlevel of the detail with which the charge distributionforces are computed considering protein II as a collec-in the proteins is modeled and the approximations usedtion of point charges in the solvent dielectric; i.e., theto represent the dielectric environment of the system.low dielectric and ion exclusion volume of protein IIIn simple models, used mostly in early simulationsare ignored. We have recently shown (28) that forces(26), a small number of charges are positioned in thein BD simulations can be computed more accuratelyproteins so as to reproduce their monopole, dipole,but with virtually the same computational cost as theand quadrupole moments, and interactions betweentest charge approximation by using effective potential-charges are computed using Coulomb’s law either withderived charges instead of test charges. To derive ‘‘ef-a constant dielectric permittivity or with a distance-fective charges,’’ a full partial atomic charge model ofdependent dielectric permittivity to implicitly modeleach protein is used to compute each protein’s electro-the dielectric variation in the system simulated. Ionicstatic potential separately by numerical solution of thestrength was modeled by Debye–Hückel screening.finite-difference Poisson–Boltzmann equation, takingMore recently, it has become possible to assign par-into account the inhomogeneous dielectric medium andtial atomic charges to all atoms in the proteins and tothe surrounding ionic solvent. Then effective chargesmodel the dielectric heterogeneity and ionic strengthare computed for each protein by fitting so that theyexplicitly by solving the Poisson–Boltzmann equationreproduce its external potential in a homogeneous di-numerically:electric environment corresponding to that of the sol-vent. To compute the forces and torques acting on pro-

Ç(e(r)Çf(r)) 0 c(r)e(r)k2f(r) Å 04pr(r). [9] tein II (I), the array of effective charges for protein II(I) is placed on the electrostatic potential grid of proteinI (II). This procedure gives a good approximation to theHere, the dielectric constant, e(r), is a function of posi-

tion, f(r) is the electrostatic potential, r(r) is the forces derived by solving the finite-difference equationin the presence of both proteins, unless the separationcharge density, k is the inverse of the Debye–Hückel

screening length (11), and c(r) defines the volume ac- of the protein surfaces is less than the diameter of thesolvent probe and the desolvation energies of thecessible to counterions; it is 0 inside the solute and the

Stern layer and 1 outside. Aqueous solvent is assigned charges in one protein due to the low dielectric cavityof the other protein become significant (28).a high relative dielectric constant of about 80, whereas



Exclusion Forces METHODShort-range repulsive forces are treated by an exclu-

sion volume prohibiting van der Waals overlap of theProcedure for Setting Up and Running BD Trajectoriesproteins. The exclusion volume is precalculated on a

The procedure for carrying out a BD simulation togrid. If a move during a time step would result in vancompute a bimolecular association rate is outlined inder Waals overlap, the BD step is usually repeated withFig. 2. The main steps follow.different random numbers until it does not cause an

overlap.a. Model Protein Coordinates

Hydrodynamic Interactions If starting from a crystal structure, it will be neces-Hydrodynamic interactions (HIs) between particles sary to add polar hydrogen atoms to the structure and

in a solvent arise because of solvent flow induced by possibly model in residues that were not defined in thethe motions of the particles. They can be incorporated electron density. Protons should be correctly positionedinto BD simulations by using a diffusion tensor [e.g., for the relevant pH and their assignment may beOseen or Rotne-Prager (29)] but their computational helped by computing the pKa values of the titratablerequirements increase rapidly with the number of hy- groups in the protein [see (38) and references therein].drodynamic centers. Their magnitude and importance The proton positions should be optimized by energyfor protein–ligand association have been studied for minimization, and perhaps a Monte-Carlo (39) or mo-simple protein models. The estimated effects on rates lecular dynamics optimization. As mentioned below,range from 15% (30–33) to 50% (34) to two-orders of even small adjustments in their positions can altermagnitude (35), and depend on the treatment of hydro- rates (36).dynamic boundary conditions and the sizes, shapes,

b. Assign Atomic Parametersand velocities of the solute molecules for which HI ef-fects were computed. The smaller value appears to hold The protein atoms should be assigned charges andfor conditions relevant for protein–protein association. radii. These can come either from molecular mechanicsHI can both increase and decrease molecular associa- force fields [e.g., OPLS (40), CHARMM (41)] or fromtion rates. For example, consider an elongated protein parameter sets derived specifically for use with a con-binding in a slitlike cleft of another protein. HI will tinuum solvent model [PARSE (42)].tend to lower the rate of translational diffusion toward

c. Compute Electrostatic Potentials for Boththe binding site. However, HI torques will favor bind-Moleculesing of the elongated protein when it is nearly aligned

with the cleft by enhancing the alignment. Thus, the Solvent and protein dielectric constants, ioniceffects of different HIs on association rates may tend strength, width of the ion exclusion (Stern) layer, andto cancel each other out. temperature of the system should be assigned. Poten-

tials should be computed on grids large enough to in-Modeling Flexibilityclude the volume over which the potential is noniso-

In simulations of protein–protein association per- tropic. If this is very large, an additional inner gridformed to date, the proteins have been treated as rigid with smaller spacing should be computed and bothbodies. The effects of flexibility have been considered grids should be used in the BD simulations. For effi-only by generating different sets of trajectories using ciency of handling, grid size should not exceed the sizedifferent protein conformations [see (36) and below]. limited by computer memory requirements; for accu-

For simulations of enzyme–substrate encounter, racy, the spacing for the inner grid should not exceedother representations of protein flexibility have been 1 Å (and smaller grid spacings are preferable).used and these could be applied to simulations of pro-tein–protein encounter: d. Compute Effective Charges

Effective charge sites are usually assigned to the po-—Reduction of atomic radii at the binding site toimplicitly mimic the flexibility of these atoms (25). sitions of the titratable residues (thereby reducing the

number of charge sites), and their magnitude and sign—Modeling the motion of particularly flexible re-gions, such as protein loops, explicitly during the BD are determined by fitting to reproduce the electrostatic

potential in a shell around the protein using a homoge-simulations. A simplified model is necessary to makethis computationally feasible. A model treating a pep- neous dielectric.tide loop as a chain of spheres, each sphere represent-

e. Choose b and q Surfacesing an amino acid residue, has been used (37). Thespheres interact via a set of forces designed to mimic The b surface should be chosen by examining the

electrostatic potentials so that at a distance b from eachthe geometry and interactions of an all-atom proteinmodel. protein, the electrostatic forces on the other protein are



isotropic and centrosymmetric. The q surface should be ies. The surface-exposed atoms of the other protein (pro-tein II) are listed. Steric overlap is defined to occur whenplaced much further away where the diffusive flux of

the molecules is centrosymmetric (see Fig. 1). Typical one of the surface-exposed atoms is projected onto agrid point with value 1 (43). This computation is muchdistances are 50–100 Å for the b surface and 100–500

Å for the q surface. Sensitivity to alteration of these quicker than a pairwise evaluation of exclusion forces,but requires the assumption that all atoms of proteinvalues should be tested during subsequent BD simula-

tions. II have the same radius as the probe used to define theprobe-accessible surface of protein I.

f. Compute Exclusion Volumes and Surface Atomsg. Define Reaction CriteriaThe exclusion volume is computed on a grid for one

protein (protein I) by assigning values of 1 to grid points The choice of reaction criteria has a large effect onrates computed and they must therefore be chosen withinside a probe-accessible surface and 0 outside. A spac-

ing of 1 Å is usually adequate, but a 0.5-Å spacing may care. Reaction criteria are required to define formationof a diffusional encounter complex. This is the complexgive better convergence and smoother, shorter trajector-

FIG. 2. Schematic diagram showing the steps in the computation of bimolecular diffusion-controlled rate constants by Brownian dynamicssimulation.



formed at the endpoint of the diffusional association protein as a hydrodynamically isotropic sphere so thatits diffusion constants are defined by sphere radius,phase from which, in the next phase of binding, thesolvent viscosity, and temperature. More sophisticatedproteins would rearrange into a tightly bound complex.estimations of diffusion constants account for proteinComplete binding does not occur in the BD simulationssurface shape (46, 47). A variable time step should bebecause of the incompleteness of the force model used.used that is smaller when the proteins are closer toOne or several reaction criteria may be monitored dur-each other and when protein II is close to the q surface.ing a trajectory. The criteria may be energetic (e.g.,The smallest time step for protein–protein simulationsbased on electrostatic energy) or geometric (rmsd ofwith rigid body proteins should typically be 0.5–1.0 ps.atoms from specified position, arrangement betweenFor efficiency, rotations may be performed less fre-protein atoms or cofactors) or based on contacts (num-quently than translations. The number of trajectoriesber of contacts, buried surface area) (see Fig. 3). A crite-run will depend on the accuracy required and the mag-rion based on the arrangement of heme cofactors hasnitude of the rates computed. The slower the rates, thebeen used for electron transfer proteins (44, 45). Moregreater the number of trajectories that will be required.generally, we have found that the energetic criterionStatistical errors in rates can be estimated either fromshould not be used and that a criterion based on forma-Poisson statistics as K01/2r100%, where K is the numbertion of two to three correct contacts performs best forof encounter events, or by subdividing the trajectories,protein–protein association (see below for details).computing rates for each subdivision, and then derivingthe error as the rmsd of the rates from their average.h. Define BD Parametersi. Run BD SimulationsRotational and translational diffusion constants

should be defined. This can be done most simply using Each simulation is started with a randomly chosenposition and orientation of protein II on the b surfacethe Stokes–Einstein relationship and treating each

FIG. 3. Schematic representation of different reaction criteria for monitoring encounter complex formation. (a) Criterion based on rmsdistance of selected atoms of protein II to their position in bound complex. (b) Criterion based on the number of residue contacts. In both(a) and (b), any number of atoms may be used and two in each protein are shown for simplicity. (c) Atomic contact criterion. See text fordetails.



and run until protein II escapes from the q surface. colleagues, is designed for BD simulation of enzyme–substrate encounter to compute steady-state and non-The number of reaction events during each simulation

is counted. Simulations can straightforwardly be run steady-state association rates. It can be used for simu-lations of protein–protein association when one proteinin parallel on multiple processors.is treated with a simplified rather than atomic-detail

j. Compute Rates and Analyze Trajectories model. It also has modules for numerical solution ofThis is done for diffusional pathways and important the finite-difference linear and full Poisson–Boltz-

regions of the proteins for diffusional encounter. mann equations, electrostatic binding free energies,stochastic dynamics, molecular mechanics energy min-

Comparison of Simulation of Protein–Protein and imization, and modeling internal protein flexibility inProtein–Small Molecule Encounters BD simulations.

The theory and the general procedure for computing —SDA (http://www.embl-heidelberg.de/ExternalInfo/association rates are the same for protein–protein in- wade/pub/soft/sda.html) was developed by us to be gen-teractions as for protein–small molecule interactions. erally applicable to the simulation of diffusional associ-However, there are some significant differences that ation of biomolecules with simple or atomic-detail mod-make the association of two macromolecules more diffi- els. Additional capabilities include a module (ECM) forcult to simulate than the association of a macromole- computation of effective charges using potential gridscule with a much smaller molecule: computed with the UHBD program and another mod-

ule to estimate electrostatic enhancement of bimolecu-—There are more degrees of freedom. Rotationallar rates by the Boltzmann factor analysis method ofdiffusion must be simulated for protein–protein as-Zhou (49).sociation, whereas small molecules can often be

treated as either a single sphere or two spheres (adumbbell) covalently bound together that are treated

EXAMPLE APPLICATION: DIFFUSIONALas separate hydrodynamic objects connected by a con-strained bond (48). ENCOUNTER OF BARNASE AND BARSTAR

—The low dielectric of both molecules must betreated; i.e., the test charge model is a worse approxi- The association of barnase, an extracellular ribo-mation for protein–protein association than protein– nuclease, with its intracellular inhibitor, barstar, pro-small molecule association. vides a particularly well-characterized example of elec-

—Reaction criteria may be harder to define as there trostatically steered diffusional encounter betweenis less likely to be a well-defined concave binding site proteins. The association rate is very fast (Ç108–109for a protein than for a small molecule. M01 s01 at 50 mM ionic strength), and studies of the

—HIs between two similar-size macromolecules are effects of mutation and variation of ionic strengthlikely to have more impact on their association rate clearly show the influence of electrostatic interactionsthan HIs between molecules of very different sizes. (7). We have shown that the ionic strength dependence

—While internal flexibility in a substrate may be of of the rate for the two wild-type proteins and the ratesnegligible importance for diffusional encounter with an for the wild-type proteins and 11 mutants at 50 mMenzyme, intramolecular motions in proteins are more ionic strength can be reproduced to within a factor of 2likely to be important. by BD simulation (36). These simulations also provided

insights into the structure of the diffusional encounterSoftware for BD Simulationscomplex in which barstar tends to be shifted from itsSeveral programs are available for performing BD position in the crystal structure of the complex towardsimulations, each with different capabilities: the guanine-binding loop on barnase.

Simulations of the association of barnase with bar-—Macrodox (http://pirn.chem.tntech.edu/macrodox.star were carried out using the SDA software followinghtml), developed by Scott Northrup and colleagues,the procedure described above. Many (10,000 per pro-may be used to perform BD simulations of the bimolec-tein variant) trajectories were generated and the con-ular association with simple and atomic-detail models.formations sampled in one trajectory are shown in Fig.It was designed particularly for performing BD simula-4. Here, we draw attention to methodological aspects,tions of the association of electron transfer heme pro-correct treatment of which is particularly important forteins. Its capabilities also include Tanford–Kirkwoodobtaining agreement with experiment.calculations to estimate pKa values and assign partial

charges and numerical solution of the linear and fullElectrostatic ModelPoisson–Boltzmann equations.

—UHBD (http://chemcca10.ucsd.edu/Çjmbriggs/ The importance of accounting for the dielectric heter-eogeneity and ion exclusion volumes of both proteinsuhbd.html) (11), developed by Andrew McCammon and



is seen by comparing the results obtained with an effec- Geometric Criteriontive charge model with those obtained with a test Two atoms on the binding face of barstar were re-charge model. As shown in Fig. 5, the use of test quired to approach within a specified rmsd, d, of theircharges to compute electrostatic forces leads to under- positions in the crystallographically observed complex,estimation of the steering forces. The experimentally as shown in Fig. 3a. This criterion could reproduce theobserved 18-fold drop in association rate on changing effects on rate for only some mutants. Moreover thisthe ionic strength from 50 to 500 mM is modeled cor- was at an rmsd d of 6.5 Å, which is too large a distancerectly with effective charges, but is modeled as only to be a realistic measure of the distance at which short-a 10-fold drop with test charges. The ionic strength range interactions between the proteins become strongdependence of the rate is most severely underestimated enough to ensure that complexation occurs subsequentat ionic strengths of 300 to 500 mM, implying that to formation of the diffusional encounter complex.the test charge interactions at these and higher ionicstrengths are very small at the encounter contact dis- Energetic Criteriontances. On the other hand, with effective charges, ionic

The electrostatic interaction energy was required tostrength dependence is reproduced from 50 to 500 mMbe more favorable than a defined threshold. This crite-ionic strength and interactions are significant at therion does not reproduce differences between mutantshigher ionic strengths. This shows that the associationwith a single threshold and is not recommended be-mechanism is the same at all salt concentrations incause it leads to an overestimate of the effects of elec-this range and that the ionic strength dependence of trostatic steering which are effectively counted twice.the rates may be ascribed solely to changes in the elec-

trostatic steering forces.Residue Contact Criterion

A defined number of correct residue–residue con-Definition of Formation of Diffusional Encounter tacts between the two proteins were required in whichComplex the distance between specified atoms in the residues

Reaction criteria based on geometry, energy, and in- was shorter than a defined length dC, as shown in Fig.termolecular contacts were tested for this system as 3b. Contact pairs were defined to be as mutually inde-

pendent as possible. Formation of two contacts shorterfollows:

FIG. 4. Trajectory simulated by BD of barstar diffusing toward its binding site on barnase. The trajectory is started with barstar placedat a random position on the sphere shown with a radius of 100 Å centered on barnase. The trajectory is about 68 ns long and the positionsof the center of the diffusing barstar are plotted at approximately 200-ps time intervals. The position of the center of barstar in the crystalstructure of the complex with barstar is indicated by the dot in the small circle. Most trajectories are shorter than the one shown and inmost, barstar does not reach the barnase binding site before diffusing away.



than 5.25 Å or three contacts shorter than 7.5 Å was listing dependent pairs of contacts, i.e., those contactsin which either atom is within dmin of an atom of thefound to give computed rates in good agreement with

experimental data and reproduce the effects of ionic other contact within the same protein. During a BDtrajectory, donor–acceptor atom contacts shorter thanstrength (see Fig. 5) (36). The distance of 5.25 Å can be

interpreted as a direct, rather than solvent-separated, dC are monitored. Then the number of independent con-tacts is computed by ensuring that no more than oneinteraction distance between atoms. A better fit for the

diverse set of mutants studied was obtained at three contact from any dependent pair of contacts is counted.With this definition, computed rates for barnase–times the experimentally measured association rate

with dCÅ 6.25 Å for two contacts (see Fig. 6), indicating barstar association agreed well with experimentalrates when two contacts shorter than dC Å 4.5 Å werethat BD is better suited to model the formation of the

transition state than the bound complex. required. Computed rates were more accurate thanthose obtained for the residue contact criterion for vari-

Atom Contact Criterion ants in which mutation affected the number of residuecontacts monitored with the residue contact criterion.This is similar to the above criterion but the atom–

atom contacts are assigned in a fully automated waySensitivity of Modelindependent of which residue each atom is in (see Fig.

The absolute magnitude of the rate computed is3c). The contacts are between hydrogen bond donor andmuch more sensitive to modifications in parameters oracceptor atoms. The specific contacts are selected soprotein structure than the relative rates for the set ofthat they are approximately equivalent in terms ofmutants or the ionic strength dependence. The im-their contribution to the bimolecular interaction en-portant effects of reaction criteria and charge model onergy. This is achieved by requiring donor–acceptorrates computed has been discussed above. Modificationpairs to be separated by distances greater than dmin (6of the following additional features of the model alteredÅ). A contact between two side chains is counted as onethe absolute rates at 50 mM ionic strength, but didcontact even if there are two or three contacts betweennot produce any significant change in the correlationdonor and acceptor atoms in the side chains. Potentialbetween computed and experimental rates for the setcontacts are defined by, first, tabulating all intermolec-of mutants studied.ular donor–acceptor atom contacts shorter than a dis-

tance dmax (5 Å) in the experimental structure of the • A factor of up to 10 decrease in rates could be ob-bimolecular complex and, second, generating a table tained by using the coordinates of the unbound barnase

from either the crystal structure or the ensemble ofstructures from the NMR determination, instead of co-ordinates from the crystal structure of the barnase–barstar complex.

FIG. 5. Ionic strength dependence of the barnase–barstar associa-tion rate: ---, Experimentally measured values (plain) and experi-mentally measured values multiplied by 3 (boldface); rrr, valuescomputed with test charges; —, values computed with effectivecharges. For defining encounter complex formation, two contactswere required with contact distances less than 6.5 Å (lower line,plain) and 7.5 Å (upper line, boldface) for test charges and 5.25 Å(lower line, plain) and 6.00 Å (upper line, boldface) for effective FIG. 6. Comparison of experimental and computed association

rates for wild-type barnase and barstar and 11 mutants. A contactcharges. Note that the commonly used test charge approximationresults in an underestimate of the ionic strength dependence and distance of 6.25 Å was required when computing the rates. The

dashed line presents the dependence log(computed rate) Å 3∗log(ex-that this can be corrected by using effective potential-derivedcharges. perimental rate).



• A factor of 2 decrease in rates could be obtained by CONCLUDING REMARKSremoving the 2-Å-thick Stern layer, or by minimizinghydrogen atom positions with the proteins separatedrather than positioned in their bimolecular complex, or Besides the barnase–barstar case described above,by using snapshots from a molecular dynamics simula- BD simulation has been used to compute protein–pro-

tein association rates in a range of applications: elec-tion of the barnase–barstar complex rather than thetron transfer heme proteins, antigen–antibody bind-crystallographic coordinates.ing, and protein self-association (17). The first• No noticeable difference was observed for this sys-applications using detailed protein models related totem when the nonlinear Poisson–Boltzmann equationthe heme protein cytochrome c and its electron transferwas used instead of the linearized equation to computepartners, cytochrome c peroxidase (44) and cytochromeelectrostatic forces.b5 (45). Electron transfer rates were computed by cou-pling models describing the electron transfer event toIn summary, the application to barnase–barstar as-diffusional encounter trajectories of the proteins. Ionicsociation shows the following:strength dependencies were reasonably reproduced,qualitative insights into the effects of mutations were

• The BD simulation method can reproduce experi- obtained, and structural information about encountermental rates well. The accuracy is such that BD simu- complexes was derived. For cytochrome c–cytochromelations should be useful for the design of mutants with c peroxidase association, two distinct binding modesaltered on-rates. were obtained, each containing many alternative con-

• In the encounter complex (transition state) for this formations, indicating a multitude of electron transferenzyme, two correct contacts are formed. Rotational orientations rather than a single dominant complex.freedom is not yet fully lost (see Fig. 7). After the simulations of cytochrome c–cytochrome c

• Some residues are more important than others for peroxidase association were performed, the crystalsteering binding. Barstar tends to bind first to residues structure (50) was solved but showed specific complexin the guanine-binding loop of barnase and their muta- formation. This apparent discrepancy may arise be-tion has a greater impact on association rates than cause the crystal structure should represent the unique

bound state, whereas the BD simulations model transi-other barnase residues in the binding interface.

FIG. 7. Encounter complexes for barnase–barstar association. Barnase and barstar are shown in boldface at their positions in the crystalstructure of their complex. The gray lines show 17 encounter positions of barstar obtained in different trajectories. The two views differ bya 907 rotation about the vertical axis. Residues 57–60 in the guanine-binding loop of barnase are indicated by circles.



2. Berg, O. G., and vonHippel, P. H. (1985) Annu. Rev. Biophys.tion state encounter complexes. A similar situation ex-Biophys. Chem. 14, 131–160.ists for the barnase–barstar system discussed in the

3. DeLisi, C. (1980) Q. Rev. Biophys. 13, 201–230.previous section.4. Szabo, A., Stolz, L., and Granzow, R. (1995) Curr. Opin. Struct.The importance of electrostatic effects has been ob-

Biol. 5, 699–705.served in BD simulations of antibody–antigen associa-5. Malmqvist, M. (1993) Curr. Opin. Immunol. 5, 282–286.tion and protein self-association. Antibody–antigen as-6. Stone, S. R., Dennis, S., and Hofsteenge, J. (1986) Biochemistrysociation rates are typically of the order of 106 M01 s01 28, 6857–6863.

at physiological ionic strength, yet BD simulations (51, 7. Schreiber, G., and Fersht, A. R. (1996) Nature Struct. Biol. 3,52) showed that the association rates of lysozyme with 427–431.the HyHEL-5 antibody are sensitive to ionic strength 8. Gosting, L. J. (1956) Adv. Protein Chem. 11, 429–554.and mutation of charged residues near the antibody’s 9. Johnstone, R. W., Andrew, S. M., Hogarth, M. P., Pietersz, G. A.,binding site. The bimolecular association rates of BPTI and McKenzie, I. F. C. (1990) Mol. Immunol. 27, 327–333.molecules have been simulated by BD (53) and com- 10. Ito, W., Yasui, H., and Kurosawa, Y. (1995) J. Mol. Biol. 248,

729–732.pared with the kinetics of disproportionation of disul-11. Madura, J. D., Briggs, J. M., Wade, R. C., Davis, M. E., Luty,fide groups on the protein surface (54). Association

B. A., Ilin, A., Antosiewicz, J., Gilson, M. K., Bagheri, B., Scott,rates areÇ107 M01 s01 at 200 mM ionic strength. TheyL. R., and McCammon, J. A. (1995) Comp. Phys. Commun. 91,increase with ionic strength, as would be expected for 57–95.

molecules of like charge, although some other proteins12. Wade, R. C. (1996) Biochem. Soc. Trans. 24, 254–259.

undergoing self-association display different ionic13. Getzoff, E. D., Cabelli, D. E., Fisher, C. L., Parge, H. E., Viezzoli,

strength dependencies (54). BD simulations were per- M. S., Banci, L., and Hallewell, R. A. (1992) Nature 358, 347–formed with several models but only with the most 351.realistic model with all atoms, and 128 charges was 14. Northrup, S. H., and Erickson, H. P. (1992) Proc. Natl. Acad.

Sci. USA 89, 3338–3342.the computed ionic strength dependence of the rate15. Einstein, A. (1905) Ann. Phys. (Leipzig) 17, 549.close to that observed experimentally. A test charge16. Smoluchowski, M. V. (1906) Ann. Phys. (Leipzig) 21, 756.model was used and internal motions were neglected,

but these approximations were expected to be partially 17. Madura, J. D., Briggs, J. M., Wade, R. C., and Gabdoulline, R. R.(1997) Encyclopedia of Computational Chemistry, in press.compensating for this system (53).

18. Ermak, D. L., and McCammon, J. A. (1978) J. Chem. Phys. 69,In the future, applications can be expected to the1352–1360.association of many other proteins including, for exam-

19. Smoluchowski, M. V. (1917) Z. Phys. Chem. 92, 129–168.ple, those involved in signal transduction, and toxin20. Northrup, S. H., Allison, S. A., and McCammon, J. A. (1984) J.proteins binding to pore proteins and enzymes. BD sim-

Chem. Phys. 80, 1517–1524.ulations of protein association are complementary to21. Huber, G. A., and Kim, S. (1996) Biophys. J. 70, 97–110.other methods to estimate the effects of electrostatic22. Zhou, H.-X. (1990) J. Phys. Chem. 94, 8794–8800.interactions on association rates such as the Boltz-23. Luty, B. A., McCammon, J. A., and Zhou, H.-X. (1992) J. Chem.mann factor analysis of Zhou (49, 55), which gives esti-

Phys. 97, 5682–5686.mates of the electrostatic enhancement of on-rates in

24. Zhou, H.-X. (1993) Biophys. J. 64, 111–1726.qualitative agreement with those obtained by BD. It is

25. Luty, B. A., Wade, R. C., Madura, J. D., Davis, M. E., Briggs,also complementary to computational methods to dock J. M., and McCammon, J. A. (1993) J. Phys. Chem. 97, 233–237.proteins and predict their bound complexes. Indeed, in 26. Northrup, S. H., Reynolds, J. C. L., Miller, C. M., Forrest, K. J.,addition to providing computed bimolecular rates, BD and Boles, J. O. (1986) J. Am. Chem. Soc. 108, 8162–8170.simulations may provide good starting structures for 27. Davis, M. E., and McCammon, J. A. (1989) J. Comp. Chem. 10,docking procedures. 386–391.

The SDA software suite for BD simulations is avail- 28. Gabdoulline, R. R., and Wade, R. C. (1996) J. Phys. Chem. 100,3868–3878.able from the authors.

29. DelaTorre, J. G., and Bloomfield, V. A. (1981) Q. Rev. Biophys.14, 81–139.

30. Antosiewicz, J., Gilson, M. K., Lee, I. H., and McCammon, J. A.ACKNOWLEDGMENT(1995) Biophys. J. 68, 62–68.

31. Wolynes, P. G., and Deutch, J. M. (1976) J. Chem. Phys. 65, 450–The authors thank Dr. Indira Shrivastava for reading the manu- 454.

script.32. Friedman, H. L. (1966) J. Phys. Chem. 70, 3931–3933.33. Allison, S. A., Srinivasan, N., McCammon, J. A., and Nortrup,

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36. Gabdoulline, R. R., and Wade, R. C. (1997) Biophys. J. 72, 1917– 45. Northrup, S. H., Thomasson, K. A., Miller, C. M., Barker, P. D.,Eltis, L. D., Guillemette, J. G., Inglis, S. C., and Mauk, A. G.1929.(1993) Biochemistry 32, 6613–6623.37. Wade, R. C., Davis, M. E., Luty, B. A., Madura, J. D., and

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